# 18

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## Number

**$18$ (eighteen)** is:

- $2 \times 3^2$

- The number of distinct pentominoes, if reflections of asymmetrical pentominoes are included in the count

- The $1$st element of the $1$st pair of integers $m$ whose values of $m \, \map \tau m$ is equal:
- $18 \times \map \tau {18} = 108 = 27 \times \map \tau {27}$

- The $2$nd abundant number after $12$:
- $1 + 2 + 3 + 6 + 9 = 21 > 18$

- The $3$rd semiperfect number after $6$, $12$:
- $18 = 3 + 6 + 9$

- The $3$rd heptagonal number after $1$, $7$:
- $18 = 1 + 7 + 11 = \dfrac {3 \paren {5 \times 3 - 3} } 2$

- The $3$rd pentagonal pyramidal number after $1$, $6$:
- $18 = 1 + 5 + 12 = \dfrac {3^2 \paren {3 + 1} } 2$

- The $5$th integer after $0$, $1$, $8$, $17$ equal to the sum of the digits of its cube:
- $18^3 = 5832$, while $5 + 8 + 3 + 2 = 18$

- Equal to the sum of the digits of its $6$th power:
- $18^6 = 34 \, 012 \, 224$, while $3 + 4 + 0 + 1 + 2 + 2 + 2 + 4 = 18$

- Equal to the sum of the digits of its $7$th power:
- $18^7 = 612 \, 220 \, 032$, while $6 + 1 + 2 + 2 + 2 + 0 + 0 + 3 + 2 = 18$

- The $6$th Lucas number after $(2)$, $1$, $3$, $4$, $7$, $11$:
- $18 = 7 + 11$

- The $8$th positive integer after $1$, $2$, $3$, $4$, $6$, $8$, $12$ such that all smaller positive integers coprime to it are prime

- The $9$th positive integer which is not the sum of $1$ or more distinct squares:
- $2$, $3$, $6$, $7$, $8$, $11$, $12$, $15$, $18$, $\ldots$

- The $10$th highly abundant number after $1$, $2$, $3$, $4$, $6$, $8$, $10$, $12$, $16$:
- $\map \sigma {18} = 39$

- The $10$th Ulam number after $1$, $2$, $3$, $4$, $6$, $8$, $11$, $13$, $16$:
- $18 = 2 + 16$

- The $10$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$ such that both $2^n$ and $5^n$ have no zeroes:
- $2^{18} = 262 \, 144$, $5^{18} = 3 \, 814 \, 697 \, 265 \, 625$

- The $11$th (strictly) positive integer after $1$, $2$, $3$, $4$, $6$, $7$, $9$, $10$, $12$, $15$ which cannot be expressed as the sum of exactly $5$ non-zero squares

- The $12$th harshad number after $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $12$:
- $18 = 2 \times 9 = 2 \times \paren {1 + 8}$

- The $13$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $9$, $10$, $11$, $17$ such that $5^n$ contains no zero in its decimal representation:
- $5^{18} = 3 \, 814 \, 697 \, 265 \, 625$

- The $15$th integer $n$ after $0$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $13$, $14$, $15$, $16$ such that $2^n$ contains no zero in its decimal representation:
- $2^{18} = 262 \, 144$

- $18 = 9 + 9$, and its reversal $81 = 9 \times 9$

- $18^3 = 5832$ and $18^4 = 104 \, 976$, using all $10$ digits from $0$ to $9$ once each between them

## Also see

*Previous ... Next*: Lucas Number

*Previous ... Next*: Abundant Number*Previous ... Next*: Semiperfect Number*Previous ... Next*: Harshad Number*Previous ... Next*: Integers such that all Coprime and Less are Prime

*Previous ... Next*: Numbers not Sum of Distinct Squares*Previous ... Next*: Integer not Expressible as Sum of 5 Non-Zero Squares

*Previous ... Next*: Powers of 2 with no Zero in Decimal Representation*Previous ... Next*: Highly Abundant Number*Previous ... Next*: Ulam Number

*Previous ... Next*: Positive Integers Equal to Sum of Digits of Cube*Previous ... Next*: Powers of 5 with no Zero in Decimal Representation

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $18$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $18$